\section{Introduction}
\label{sec:introduction}

Half Space Proximal (HSP) is an algorithm for extracting a directed or undirected graph out of a point cloud. The algorithm for a three-dimensional cloud is as follows:
Let $L_1$ be a list of points in space, let $x\in L_1$, let the set of points that will be the neighbors of $x$ be $L_2=\emptyset$ and let $F=\emptyset$ be the points in the forbidden area. We shall repeat the following steps until $L_1\backslash F=\emptyset$.
\begin{enumerate}
\item Find $y\in L_1\backslash F$ such that $d(x,y)$ is minimum.
\item We shall add to $F$ the points that are in the open half-space containting $y$ that is determined by the plane perpendicular to the segment $x,y$ and cuts the segment in half.
\item We add $y$ to $L_2$.
\end{enumerate}

All the points contained in $L_2$ are points the that will be connected to $x$ in the resulting graph. $|L_2|$ is called the out-degree for the point $x$. The out degree for the cloud $L_1$ is the maximum out-degree for any point in $L_1$.

We shall give a set $L_1$ of points in three-dimensional space in which a given point $x\in L_1$ has the greatest possible out-degree, hence proving that the out degree for any set of points in three-dimensional space is less or equal than the out degree for our point $x$. To accomplish this, we are going to build the cloud point by point, thus finding the out-degree for HSP-3.
